Because I'm primarily interested in this question from the point of view of $\infty$-categories (in this case, modeled by quasicategories), I'll ask this question using that terminology. In particular, I'll just say "category" instead of $\infty$-category or quasicategory, to keep things legible. I suspect that, up to adding in the word "homotopy," a lot of what I'm going to say holds very generally.
It's not hard to work out that if $Top$ is the category of $\infty$-groupoids (or anima, according to Peter Scholze), then there is an equivalence between comodules over any $X\in Top$, with respect to the Cartesian monoidal structure, and the slice category $Top/X$. I'll write $Comod_X(Top)\simeq Top/X$. Furthermore, by Lurie's straightening/unstraightening (a.k.a. the $\infty$-categorical Grothendieck construction), we have an equivalence $Top/X\simeq Fun(X,Top)$. The first category, $Comod_X(Top)$, is clearly comonadic over $Top$ via the forgetful functor $U\colon Comod_X(Top)→Top$. Moreover, the colimit functor $Fun(X,Top)→Top$ factors through $Comod_X(Top)$ and is the left adjoint of an adjoint equivalence realizing the composite equivalence $Comod_X(Top)?Fun(X,Top)$. Thus, $Fun(X,Top)$ is comonadic over $Top$ via the colimit functor $colim\colon Fun(X,Top)→Top$.
My question is whether or not this is more generally true, perhaps for some generic abstract reasons that I'm not aware or, or not seeing. Note that it is not in general true (I don't think) that $colim:Fun(D,C)→C$ is comonadic for arbitrary categories $D$ and $C$. However, I'm happy with assuming that $D$ is an $∞$-groupoid and that $C$ is "nice," i.e. at least (locally) presentable.
So really the question is, are there check-able conditions under which the colimit functor $colim:Fun(X,C)→C$ is comonadic, under the assumptions that $X$ is an $∞$-groupoid and $C$ is presentable?